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178       XIII    INTEGRATION

linear isometry Up of L(X, JJL) onto L(Y, v) which extends /. (Show that U has a
unique extension to a linear bijection of the space E(X, p) of classes of /x-integrable
step functions on X, onto the space E(Y, v) of classes of v-integrable step functions
on Y. Begin by proving that if inf(A, S) = $9 then also inf(/(A), C/(B)) = <f> and
Z7(sup(A, S)) = sup(tf(A), t/(B)); then deduce that U(A) <i t/(B) whenever A <^8;
finally, show that for arbitrary A, S we have /(inf(A, S)) = inf(C/(A), t/(S)) and
t/(sup(A, B)) = sup(t/(A), tf(8)). Observe then that Np(<7(/)) = Np(/) for/e E(X5 /*).)
Show that the image of Lg(X, jit) n L(X, ^) under C/p is LJ(Y, v) n L(Y, v),and
that the restriction of Up to L(X, ju,) n L^X, ft) is an isomorphism of Banach
algebras.

(d) Conversely, for a p such that 1 ^p< -foo, let V be a linear isometry of
L(X, jit) onto L(Y, v) such that the restriction of V to LJ(X, fi) n L(X, ^) is an
algebra isomorphism onto Lg(Y, v) n Lg(Y, v). Then the restriction of Fto I(X, p,)
is an isometry onto I(Y, v) which maps f to $.

9. Let X be a compact space such that //,(X) = 1. Let % be a set of integrable subsets
of X, such that the relations A e % and B e % imply that X  A e X and A n B e %.
Suppose also that the set {A: A e } is dense in the metric space I(X, fji) (Problem 8).

(a)    Let (Cj)i^j^ be a finite partition of X into /x-integrable subsets. Show that
for each >0 there exists a partition (A^i^j^n consisting of sets in X such that
/x(D(Q, A,)) <; e for 1 <,j < n. (Observe that if ju,(D(C,, B.,)) <j 8 for 1 gy ^ n - 1,
then jU,(Bj n Bk) < 28 for 1 <;/ < k :g   1; if N is the union of the By n Bfc, consider

the sets A, = B, -N (1 ^/ <i TZ - 1) and An = X -"Q A,).

(b)    Deduce from (a) that for each finite partition y = (Q)i <J<n of X into integrable
subsets, and each e>0, there exists a finite partition a==(Aj)i^j^n consisting of
sets in % and such that (in the notation of Section 13.9, Problem 27)

H(a/y) + H(y/a) ^ fi.

(Reduce to the case where none of the Cj is negligible, and use the fact that the function
r-> t log r is zero and continuous at t = 1.)

10. Let X be a compact space such that ju,(X) = 1, and let u; X-X be a /^-measurable
mapping such that u(^)  jit.

(a)    Let a be a finite partition of X into integrable subsets, and let % be the set of

n- 1

all finite unions of subsets of X belonging to one or other of the partitions V u~J(oc).

J = 0

Suppose that the set of classes {A: A e X} is dense in I(X, ft) (Problem 8). Show that
we then have h(u) = h(u, a). (It is enough to show that h(u, /?) <[ h(u, a) for every
finite partition J3 of X into integrable subsets; observe that we have

V u-J(oc)\ +H(B/ \) u-J(o
J = o            ]          \  I J~o

and use Problem 28(d) of Section 13.9, and Problem 9 above.)

(b)    Under the hypotheses of (a), show that if in addition u is bijective and u""1 is
jit-measurable, then h(u)  0. (Observe that the set of classes {(zrXA))"; A e%] is
again dense in I(X, ft), and use Problem 28(c) of Section 13.9.)

(c)    Suppose that u is bijective and u~~l is /x-measurable. Let ' be the set of finite

unions of subsets of X belonging to one or other of the partitions V  J(a), and supposee the graph of w, and